Theoretical Output Current Spectra of Three Phase Current Source Converters

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1 Thorcal Oupu Currn Spcra of Thr Phas Currn Sourc Convrrs chal Brhoff, Frrch W. Fuchs Chrsan-Albrchs-Unvrs of Kl Kasrsr., Kl, Grman Tl Sabn Pschk Howalswrk-Dusch Wrf GmbH Wrfsr. /, Kl, Grman Tl Kwors «Currn Sourc Invrr CSI», «Harmoncs», «Puls Wh oulaon PW», «Passv Flr» Absrac Th gomrc-wall-mol b Black has succssfull bn appl o calcula h harmonc spcra of volag sourc convrrs. Ths papr rvs h applcaon of h gomrc-wall-mol on h oupu currns of a currn sourc convrr, CSC, b ulzng ual consraons o volag sourc convrrs o rmn h spcra of h puls oupu currns occurrng n a CSC. Inroucon Powr qual usuall s rmn b h phas splacmn of volag an currn funamnal as wll as h currn an volag soron. Th qual of h currn wavm s commonl masur b h oal harmonc soron facor o valua h harmonc conn. Furhrmor spcfc rangs of h currn spcra ar arss b EI concrns. Hnc currn harmoncs hav gan spcal anon powr qual rqurmns n rcn ars. Thus flr rang s an nvabl par of powr convrr sgn. To m boh rqurmns o flr sgn, h conomcal as wll as powr qual aspcs, currn harmoncs o b xpc b an puls wh moulaon srag n combnaon wh a spcfc flr opolog hav o b prc an h complanc o oblgaor sanars has o b chck. Ths am has bn subc of man publcaons an spcall h currn harmoncs of h mos common hr phas volag sourc convrr hav bn analz xnsvl. In m oman s possbl o calcula h phas currn harmonc conn as a funcon of h moulaon n wh approxmaons whch l smpl quaons ach moulaon schm an frs orr flrs a nglgbl rror []- []. Wh approachs n h frqunc oman accura bu n urn also complca rsuls can b achv []-[8]. Th lar soluons ar mor comprhnsv as h rnr nmaon no onl abou h roo man squar valu of h sum of all harmoncs bu also h compl phas currn spcra xampl b mans of applng h gomrc-wall-mol approach b Black [9]. Thus s capabl o suppor analss of an moulaon mho an spcall an flr opolog as rsulng calculaons ar as o hanl n h frqunc oman. Bas on hs xprncs a novl analss s prsn o rv h phas currn spcra of a currn sourc convrr, CSC. Th Fourr analss accorng o Black s gomrc-wall-mol s combn wh ual consraons rgarng hr phas volag an currn sourc convrrs [], []. Th cohrnc bwn a slcon of spac vcor squncs h currn sourc convrr an wll known moulaon wavms bng sanar applcaon o h volag sourc convrr s pon ou []. orovr h horcal oucoms of hs analss hav bn vrf b xprmnal rsuls. In h frs scon ual ssus rgarng currn sourc an volag sourc convrrs ar xplan. Ths s follow b h scon scon whch als wh h vlopmn of h oubl Fourr srs

2 accorng o Black [9] an s applcaon on h CSC. Afr clarfcaon of h funamnals calculaon rsuls an corrsponng xprmnal oucoms ar prsn h oupu currn spcra of a CSC. Fnall an assssmn h CSC as compar wh h VSC s gvn whch s follow b a concluson. Dual of VSC an CSC convrrs Snc mhos o rmn h oupu volag spcra of a wo lvl hr phas VSC ar wll known s comabl o appl such xprncs on h CSC b ulzng ual ruls. As scrb n [] an [] all planar lcrcal nworks can b ransm no ual ons. Bu snc h acual hr phas convrr opologs comprs non planar graphs h compl VSC opolog canno b ransm no h CSC opolog rcl. Howvr hs s no ncssar h purposs ha ar pursu n hs papr. B rgarng h VSC an CSC opologs n a smplf mannr as shown n fgur, a ransmaon bcoms possbl an spcall rasonabl furhr consraons of ual n puls parn. Th ransmaon ha was xcu n fgur s complan wh h gnral ual ruls [], [], sang ha nos woul b convr o loops an b hs w conncons woul b convr no la an vc vrsa. Volags bcom currns an srs conncons woul b rplac b paralll conncons o nam onl h mos mporan opraons of ual ransmaon. V Z V V / Z V Dual Transmaon Z V Z V V / Z V ' Z V Y Y Y I Y Y Y Y Y Y Sar-Dla Transmaon ' ' ' ' ' Fg. : Acual crcu opologs h VSC an CSC wh smplfcaons ransmaon an hr corrsponng graphs

3 B obsrvng h lgh color pas of fgur h bnfs of hs consraons bcom obvous: Th nsananous valus of h volag sourcs onl can assum quans of V n ±V /, h nsananous valus of h currn sourcs woul b corrsponngl ±I /. As h volag sourcs V - V can b rgar proporonal o h currn sourcs, an wh h facor V /I a rc assocaon of h moulaon schm boh convrr ps can b sablsh. B rvng PW parns h CSC from alra wll known an analz PW schms h VSC, s fasbl o appl h rsuls ha wr foun n [] [9] on h CSC wh onl ll f as h rsulng phas currns ar h sum of h ncomng currns n nos, an n h rgh mos crcu agram of fgur. Th mos ffcul par rmanng s o assgn a spcfc moulaon an carrr wavm rspcvl o on unqu swchng squnc of spac vcors h CSC. Ths has bn on whn h scop of hs work som moulaon srags, s abl I. Th frs row splas h swchng squnc of spac vcors n h rspcv scors of h spac vcor plan urng on swchng carrr pro. Th las wo rows rprsn h corrsponng moulaon an carrr wavms. Th rlaon bwn spac vcor moulaon an carrr bas moulaon s rv n [], []. Tabl I: Cohrnc of CSC -PW an carrr bas PW.8 oulaon oulaon oulaon I I I Spac vcor squnc I I,,,,,,,,,,,,,,,,,, I I I I,,,,,,,,,,,,,,,,,,,,,,,, I I I I,,,,,,,,,,,,,,,,,,,,,,,, I I I I I Corrsponng moulaon wavm Carrr wavm saw ooh, ralng g rangl rangl

4 oulaon mho rsmbls an opporun o ralz h las swchng opraons. Ohr squncs of h acv spac vcors ar concvabl an coul b ralz b h moulaon wavms corrsponng o PW mhos o an o xampl. Th acual nflunc on h numbr of swchng opraons s caus b h carrr wavm. oulaon mhos an consu wo PW mhos ha woul caus qual swchng losss as moulaon spcfc oprang pons onl, ha s ϕ / moulaon mho an ϕ -/ moulaon mho []. Ths ffc coms along wh h bnf of lss rppl n h flr phas currn u o h ncras numbr of swchng opraons. Th gomrc-wall-mol Inall h gomrc-wall-mol was appl h analss of asnchronous snusoal moulaon schms o ovrcom h problms caus b non proc bhavor of h puls wavm [] - [8]. A wo mnsonal vw h acuall hr mnsonal consr mol s gvn b fgur. Th walls ar rprsn b h bounars of h sha aras wh a hgh of V or I rspcvl pnng on h convrr p. Th walls ar orn prpncular o h x- plan. B h assumpon of unpolar swchng a c componn A s nrouc whch can b srgar n furhr calculaons. Th nrscons of h ln x/p wh h moulaon wavms / n ± h bounars of h sha aras nca a puls ranson of V or I pnng on h convrr opolog ha s obsrv. Th rsulng bpolar puls ran whou c componn s a mach of h volag or currn ouln lvr b h sourcs of h corrsponng lgh color crcu agrams of fgur. Th unpolar puls racor can b xprss b a oubl Fourr srs wh h complx Fourr coffcns rmn b. Hr p s h carrr rao p c /, c s h angular carrr frqunc an s h angular frqunc of h moulaon wavm s funamnal. For llusraon purposs h mos smpl cas was chosn: a snusoal moulaon wavm along wh naural samplng. Th moulaon wavms ar smmrcall arrang aroun lns of mulpls of o ralz a mol ha suppors a rangular carrr wav shap. For a al xplcaon of h nroucon of rgular samplng an ohr moulaon funcons s []-[8]. x/p V, V,I x c Fg. : Th gomrc wall mol A F x, A B [ A n B sn n ] [ A B sn ] ± [ A n B sn n ] m n± n n F x, n n m m m

5 Equaon can b aap h rmnaon of h Fourr coffcns of h puls oupu quans of h sourcs pc n h lgh color crcu agrams of fgur. Thr woul assum a m lk whn a rangl carrr s concrn an alrnavl woul b h appropra xprsson whn a saw ooh carrr wh a ralng g s assum whr ncas h moulaon wavm. Hr h moulaon n vars whn h rang of... Th ovrmoulaon rgon s no rgar an naural samplng s assum whn applng hs xprssons. ; I V V n n ; I V V n n In h parcular cas of h moulaon mhos gvn b abl I h calculaons wr carr ou as follows. Wh moulaon funcon xprss b quaon an wavms an xprss b follows ha quaon can b mplo n o fn h soluon moulaon mho an quaons can b nsr no o fn h corrsponng soluon xprssons. To calcula h rsulng ac oupu volags of h VSC all common mo volag porons ha o b subrac from h alra rmn spcra [8]. On h ohr han h CSC h rsulng currn harmoncs n phas on nsanc can b rmn b 7 h cas ha moulaon mho s appl. Th calculaon of h phas currn spcra PW mhos an ha o b conuc accorngl. or or or - - I n n 7

6 Calculaon rsuls Calculaons of h oupu currn harmoncs wr carr ou b mans of h gomrc-wall-mol accorng o all hr moulaon srags along wh h corrsponng carrr wavm ach as ls b abl I. Naural samplng was assum bu h calculaons can also b aus rgular samplng accorng o [], [8]. Equaon was numrcall solv usng ahca T bu h analcal soluon all moulaon wavms gvn b abl I can b look up n [8]. A carrr rao of p was assum an h s ban squncs wr confn o lmns ach. Th oucoms ar prsn n h agrams of fgur, whr h pak valus of h CSC oupu currn harmoncs ν î ar sanarz wh h c currn I an h frqunc f s scal wh mulpls of h carrr frqunc f c. ν î /I ν î /I Fg. : Calculaon rsuls: sanarz oupu currn spcra vrsus sanarz frqunc an moulaon n, lf: o, rgh: o an o cra h sam oupu currn spcra ν î /I ν î /I Fg. : asurmn rsuls: sanarz oupu currn spcra vrsus sanarz frqunc an moulaon n, lf: o, rgh: o an o cra h sam oupu currn spcra

7 Th spp ouln of h graphs as appars vrsus h moulaon n rsuls from h calculaons bng conuc scr oprang pons wh,.,..,. As moulaon srags o an o l h sam harmonc prmanc h ar rprsn b on agram onl n fgur. Th comparson of h harmonc prmanc of all hr ffrn moulaon srags rnrs h cran of wha was assum. Obvousl u o h av swchng opraon h moulaon srags o an hr follows a harmonc prmanc ha s supror o ha of o. Ths can b nca b h lowr frqunc harmoncs bng mor omnan o han h ohr wo moulaon mhos. asurmn rsuls asurmns wr akn from an IGBT currn sourc convrr accorng o h smplf quvaln crcu of fgur. I was opra wh a c currn of I A. Th hr phas loa consss of an Cflr. mh, C 8 µf an an ausabl loa rssanc. Th swchng carrr frqunc was s o f c khz. Th convrr s oupu currn was masur b a currn prob a sa sa opraon. asurmn valus wr sampl an plo wh an ONO SOKKI CF- FFT analzr. Th aa valuaon nclung h Fourr analss was on usng alab T. Fgur rvals h rsuls of h masurmn squncs as rcl corrsponng o fgur. Thr ar onl small vaons bwn analcal an xprmnal rsuls. Rmanng vaons bwn masurmns an hor ar suppos o orgna from horcal assumpons ha ffr o h acual prmss such as currn prob naccurac or prsumpon of naural samplng nsa of h acuall appl rgular samplng. Th lar problm can asl b a b h xnson of h prsn calculus rgular samplng [], [8]. Also a cran amoun of c currn rppl an rsonan ffcs of h flr ha ar gnor b h calculaons ar lkl o occur n xprmn. Comparson wh volag sourc convrrs Th quson of a comparson rgarng h harmonc prmanc of currn sourc an volag sourc convrrs s nvabl. A unf moulaon mho prov h followng can b sa. Whn obsrvng fgur s vn ha all harmoncs gnra b h currn sourc convrr as sanarz wh h c currn ν î /I ar hghr han h sanarz volag harmoncs ν û /U rnr b a volag sourc convrr b xacl h facor. Hnc a monsraon of h oupu harmoncs of volag an currn sourc convrrs as sanarz wh h funamnal woul b prfrabl. Such graph woul b val boh convrr opologs. Tabl II rvals h corrsponng analcal rsuls. Hr as an xampl h wll known moulaon funcon cra b qual ulzaon of h wo ffrn zro volag spac vcors as commonl us h VSC s also frs colu, scon row of abl I s combn wh boh, a saw ooh an a rangular carrr. A unf fnon of h moulaon n s prsum gvng a rang of.. ovrmoulaon s srgar boh convrr ps. Th lowr wo rows rprsn h swch valus of boh convrr opologs urng on xmplar swchng carrr pro. Th avanag of h scon colu moulaon srag volag sourc convrrs now bcoms obvous: Th favorabl oupu volag spcra achv wh a rangular carrr wavm woul b ralz b h sam numbr of swchng opraons as h wors cas rgarng harmoncs gnraon ha woul b a saw ooh carrr. On h ohr han concrnng h currn sourc convrr s vn ha as soon as h carrr wavm vas from a saw ooh h swchng opraons woul b oubl an so h swchng losss.

8 Tabl II: Comparson of swchng opraons VSC an CSC Saw ooh carrr, ralng g Trangular carrr Harmonc spcra Vˆ Vˆ ˆ ˆ ν ν Vˆ Vˆ ˆ ˆ ν ν Swch valus VSC V V V V / -V / V / -V / V / -V / I V V V V / -V / V / -V / V / -V / I CSC I I -I -I Concluson As mnsonng a powr lcroncs ssm an assssmn of h suggs flr or moulaon schm s nvabl. Espcall h rmnaon of h xpc currn soron s an mporan ask. To cop wh hs challng a novl approach h currn sourc convrr has bn prsn. Th wll known gomrc-wall-mol coul succssfull b appl on h currn sourc convrr b rgarng uals wh h volag sourc convrr. Thrb h analcal rmnaon of h phas currn spcra of hr phas currn sourc convrrs has bn ralz. B usng hs mho on h CSC wh an gvn moulaon mho or spcall an flr crcur h phas currn spcra can b rmn analcall a an gvn moulaon n. Th analcal rsuls agr wh xprmnal ons ha wr sampl hr ffrn parcular PW mhos. Fnall wh h labora rsuls prov a brf assssmn of h CSC oupu currn spcra as compar o h VSC oupu volag spcra was gvn.

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